TechnologyUK - Mathematics - Trigonometry - The Cosine Function The cosine function Trigonometric identities The Cosine Function Although the sine function is probably the most familiar of the six modern trigonometric functions, the cosine function comes a close second. Indeed, the sine and cosine functions are very closely related, as we shall see (if you are not familiar with the sine function, you may wish to read the page entitled "The Sine Function"). However, whereas the modern sine function is closely related to the chord function used by the ancient Greeks, the cosine function has no equivalent in Greek trigonometry. In fact, the Greeks had no other trigonometric functions apart from the chord. Nevertheless, they were aware of the association between the sine and the cosine functions, albeit not in a trigonometric sense. The way in which the two functions relate to each other is captured in geometric terms by the Pythagorean theorem, which defines the relationship between the sides of a right-angled triangle. Consider the illustration below. The relationship between sine and cosine You should be able to see from the diagram that the half-chord AD is one leg of the right-angled triangle ADC.

The other leg is line segment CD, which lies on the perpendicular bisector of chord AB. The hypotenuse AC of the right-angled triangle is a radius of the circle. The illustration shows a unit circle, which has a radius of one unit in length. The length of the half-chord subtending central angle θ will therefore have the same value as the sine of angle θ. Similarly, the length of the line segment joining the chord's mid-point to the centre of the circle will have the same value as the cosine of angle θ. Note that for a circle of arbitrary radius r, the length of line segment CD would be equal to cos (θ ) r. Just in case you are still a little uncertain about how all this works, let's revisit how the sine and cosine functions are defined in terms of a right-angled triangle. Consider the right-angled triangle shown below. In any right-angled triangle, one of the internal angles will always be ninety degrees (90°). Since the internal angles in any triangle must always sum to one hundred and eighty degrees (180°), the other two internal angles must both be acute angles (i.e. less than ninety degrees).

The side opposite the right-angle is always the longest side, and is called the hypotenuse. In relation to either one of the acute angles, the side opposite the angle is referred to as the opposite. The remaining side (the side common to both the angle in which we are interested and the right angle) is called the adjacent. In triangle ABC, sin (θ ) = a/c and cos (θ ) = b/c In triangle ABC, we are interested in angle θ. We have used the convention of labelling each vertex with an upper case character, and the side opposite each vertex with the corresponding lower-case character. The opposite in relation to angle θ is side a. The adjacent is side b, and the hypotenuse is side c. The sine of an angle in a right-angled triangle is defined as the quotient of the lengths of the opposite and the hypotenuse. Thus, in triangle ABC, the sine of angle θ is given as: sin (θ ) = a c The cosine of an angle in a right-angled triangle is defined as the quotient of the lengths of the adjacent and the hypotenuse.

Thus, in triangle ABC, the cosine of angle θ is given as: cos (θ ) = b c In the illustration below, we show part of a unit circle with its centre at point A and a radius r = 1. We have extended line segment BC to a point D on the perimeter of the circle, so that line segment BD becomes a chord of the circle. In the right-angled triangle ABC the vertical leg, i.e. the half-cord subtending angle θ, represents the sine of angle θ. The horizontal leg represents the cosine of angle θ. You have probably also noted the fact that, with respect to angle θ, the legs corresponding to the sine and cosine are also the opposite and adjacent sides of the right-angled triangle respectively. The sine and cosine are line segments of the unit circle Look at this triangle, and think about what we know from Pythagoras' theorem. In a right-angled triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides. In this case, since the length of the hypotenuse is one (1), then the square of the hypotenuse will also be one.

From this, we can reason that the sum of the squares of the sine and cosine must be equal to one. Now, bear in mind that for any similar triangle (i.e. a triangle in which all of the angles are the same and the lengths of the sides are in the same proportion to one another) the sine and cosine values will not change, since they relate to the ratios of the sides to one another regardless of their lengths. Thus, from the Pythagorean theorem, we derive the following relationship between the sine and cosine of an angle θ: sin2 (θ ) + cos2 (θ ) = 1 From the above equation, and looking at the representation of the sine and cosine as line segments in the unit circle, we can derive some other interesting facts about the relationship between the sine and cosine.

First of all, we can re-arrange the equation in order to derive an expression for the sine of an angle in terms of its cosine, and vice versa: sin2 (θ ) = 1 - cos2 (θ ) cos2 (θ ) = 1 - sin2 (θ ) Now look at the diagram again and consider how things would change if, instead of angle θ, we were interested in finding the sine and cosine of the other acute angle, i.e. angle ABC. Line segment AC now becomes the opposite, and by definition the sine of angle ABC. Line segment BC meanwhile becomes the adjacent, and by definition the cosine of angle ABC. Since the two acute angles in a right-angled triangle are complementary (i.e. they must add up to ninety degrees), we can derive the following relationships between the sine and cosine of angle θ: sin (θ ) = cos (90° - θ ) cos (θ ) = sin (90° - θ ) Hence the sine of an angle is equal to the cosine of that angle's complement. For this reason, the cosine function is considered to be a co-function of the sine function (which is probably why it is called a cosine).

If you have read the page entitled "The Sine Function", you may recall that each of the three main trigonometric functions (sine, cosine and tangent) is defined as the ratio of two of the sides of a right-angled triangle. As we saw above, the sides are labelled in accordance with the way in which they relate to the acute angle in which we are interested, and the trick is to remember which of the sides to use in order to calculate the value of whatever trigonometric function we want to find. So, just as a reminder, here once again is the (relatively) easy-to-remember mnemonic SOH-CAH-TOA, which should remind you that: Sine = Opposite over Hypotenuse Cosine = Adjacent over Hypotenuse Tangent = Opposite over Adjacent Thankfully, the days of having to use trigonometric tables to find the cosine of an angle (or the angle corresponding to a given cosine value) are long gone. Modern calculators can instantly perform the necessary calculations, providing you know how to use the calculator correctly.

Nevertheless, for those interested in such things, we have used Microsoft Excel to produce our own table of cosines for angles ranging from zero degrees (0°) up to three hundred and sixty degrees (360°) in increments of one tenth of a degree. To see the table, click here . Finding the cosine of an angle with the built-in calculator provided in Microsoft Windows is relatively straightforward. Let's assume we want to find the cosine of an angle of seventy-five degrees (75°). If you have it, start the Microsoft Windows built-in calculator by clicking on the start button in Windows 7 and selecting All Programs>Accessories>Calculator (other versions of Windows might require a different sequence of keystrokes). Select the scientific version of the calculator from the application's View menu. To find the cosine of seventy-five degrees using the Windows calculator, enter the following keystrokes (if you don't have the Windows calculator, use any available calculator that can perform trig functions): Enter these keystrokes to find the cosine of seventy-five degrees Make sure that you have the calculator set to the Degrees mode (unless you are planning to enter a value for the angle in radians or gradians).

If you have entered the keystrokes correctly, you should now see the following display: The calculator displays the value of cos (75°) Now let's assume that you want to find the size of an acute angle in a right-angled triangle by first finding the quotient of the adjacent and hypotenuse (i.e. the angle's cosine). Providing you have values for the lengths of the sides of the triangle, you can find the value of the angle's cosine easily enough. Finding the angle is then simply a case of applying the arccosine function (which is the inverse of the cosine function) to the result. Let's find the value of angle θ for the right-angled triangle illustrated below. We want to find the size of angle θ For angle θ, sides b and c are the adjacent and the hypotenuse respectively. We can actually find the size of the angle in a single operation on the Windows calculator. We do this by using brackets in the key sequence entered, in order to get the calculator to find the quotient of the adjacent and hypotenuse (i.e. the cosine of the angle) first. We then apply the arccosine function to the result.

Here is the key sequence to use: Enter these keystrokes to find the size of angle θ If you have entered the keystrokes correctly, you should see the following display: The calculator displays the value of angle θ As with the sine function, we can apply the cosine function to angles of greater than ninety degrees. To see an interactive demonstration of how the cosine varies as the size of the angle changes, click here (note: your browser must support Java in order for the interactive page to work). The demonstration uses the unit circle to demonstrate how the cosine values vary with the angle of rotation. Note that the cosine is represented by the horizontal leg of a right-angled triangle. The vertical leg is the half-chord subtending the central angle in which we are interested, and also represents the sine. The hypotenuse is a radius of the circle, and is also the terminal side of the angle of rotation (the initial side being the positive x-axis). As the angle of rotation increases beyond ninety degrees, the right-angled triangle moves into a new quadrant. The series of screen shots below illustrate the point (we have added the quadrant numbers to the screen shots for clarity).

The unit circle has its centre (shown here as point O) at the origin of a Cartesian coordinate system, and a radius of one unit. The length of the vertical leg of the right-angled triangle at any point is therefore equal to the absolute value of the sine of the central angle subtended by the half-chord. Similarly, the length of the horizontal leg of the triangle is equal to the absolute value of the cosine of the central angle subtended by the half-chord. The cosine in quadrant I The cosine in quadrant II The cosine in quadrant III The cosine in quadrant IV The angle of rotation in the first quadrant is forty-five degrees (45°), and is therefore the same as the angle subtended by the half-chord. In the second screenshot, we have increased the angle of rotation by ninety degrees to one hundred and thirty-five degrees (135°). The half-chord is now in the second quadrant, but the central angle subtended by the half-chord is once again forty-five degrees. We see the same thing in the third and fourth quadrants.

The angle of rotation continues to increase as we move point P anti-clockwise around the circumference of the circle, but the central angle subtended by the half-chord in each quadrant will never exceed ninety degrees. Both the sine and cosine trigonometric functions can be defined in terms of a point lying on the circumference of the unit circle. That point is the point at which the half-chord touches the circumference of the unit circle (point P on the diagrams shown above). The angle θ to which the cosine corresponds is the angle between the positive x-axis and the radius joining point P and the centre of the circle O, measured anti-clockwise. The cosine of this angle is defined to be the x-coordinate of point P. We can express this simply as: cos (θ ) = x Like the sine function, the cosine function has values in the range one to minus one (1 to -1). In the first and fourth quadrants (the quadrants to the right of the y-axis), those values will always be between zero and one, since x will always be equal to or greater than zero.

In the second and third quadrants, (the two quadrants to the left of the y-axis), the values will always be between zero and minus one, since x will always be equal to or less than zero. The cosines of the angles zero degrees (0°), ninety degrees (90°), one hundred and eighty degrees (180°) and two hundred and seventy degrees (270°) will be one (1), zero (0), minus one (-1) and zero (0) respectively. We saw above that in terms of the unit circle, the x and y coordinates of any point P on the circumference of the circle are given by the cosine and sine respectively of the angle between the positive x-axis and the line segment connecting the centre of the circle to point P, measured anti-clockwise. As we have seen, this line segment forms the hypotenuse of a right-angled triangle for which the cosine gives the length of the horizontal leg (sometimes referred to as the run), and the sine gives the length of the vertical leg (sometimes referred to as the rise). The tangent function, which we talk about elsewhere, gives the slope (i.e. the quotient of the rise and the run).

Both the sine and cosine functions are said to be periodic functions with a period of three hundred and sixty degrees (360°) or 2π, because the values returned by each function will be repeated after each complete cycle. This means that we can find the sine or cosine of any angle, no matter how large. A rotation of two and a half turns, for example, is equivalent to an angle of nine hundred degrees (900°). The sine and cosine would be given as: sin (900°) = sin (900 (mod 360)°) = sin (180°) = 0 cos (900°) = cos (900 (mod 360)°) = cos (180°) = -1 The mod operator used here is short for modulus, and yields the remainder produced when one number is divided by another. In the example used here, we are dividing nine hundred by three hundred and sixty, which produces a remainder of one hundred and eighty. The periodic nature of the sine and cosine functions can be demonstrated graphically by plotting a graph of these functions against a range of angles, as illustrated below.

Like the sine function, the cosine function is often used to model the behaviour of cyclic phenomena. A cosine wave represents a smooth, repetitive oscillation in exactly the same way that a sine wave does, except that (in terms of the unit circle) it is a graph of the way in which the value of x varies over time, rather than y. A graph of sine and cosine values plotted against angles from 0° to 720° You can see from the graph that both the sine and cosine functions go through a complete cycle at three hundred and sixty degree intervals. You may also observe that the cosine wave is ninety degrees (90°) in advance of the sine wave. The cosine wave is thus said to lead the sine wave by ninety degrees. Apart from this obvious difference, the cosine wave is identical in shape to the sine wave and, like the sine wave, is often used to model the behaviour of cyclic phenomena.

In its most basic form, a cosine wave is a function of time that is described by the formula: x (t) = A · cos (ωt + φ) where: A = the amplitude of the function (the maximum absolute value of y) ω = the angular frequency (usually measured in in radians per second) t = the elapsed time φ = the phase, i.e. the angle (in radians) when t = 0 The history of the cosine function is closely bound to that of the sine function. Indian mathematicians had developed the cosine function by the fifth century CE, which they used together with the sine function in the context of performing astronomical calculations. Their knowledge of the sine and cosine functions, along with the rest of their astronomical and mathematical knowledge, was subsequently acquired by Arab scholars as the Islamic empire expanded. Much of the work of these Arab scholars, translated into Latin and accompanied by the extensive trigonometric tables they had compiled, had found its way into Europe by the twelfth century CE. By the end of the sixteenth century CE, European scholars had produced trigonometric tables which were accurate enough to be used for astronomical calculations even as recently as the early twentieth century.

Somewhat more recent developments have included the development of various representations of trigonometric functions as infinite series. Without dwelling too much on the details, suffice it to say that values for the sine or cosine of many angles cannot be calculated using simple algebraic formulae. As with the sine function, the cosine function of an angle can only be accurately expressed as the result of adding together an infinite number of terms. One such representation gives the cosine of an angle x (expressed in radians) as: cos (x) = 1 - What Is Trigonometry? What Is Trigonometry? By Robert Coolman, Live Science Contributor | May 30, 2015 04:58am ET MORE Sine, cosine and tangent are the principle trigonometric functions. Credit: YuryZap | Shutterstock Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles. Further still, trigonometry has astoundingly intricate relationships to other branches of mathematics, in particular complex numbers, infinite series, logarithms and calculus.

The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle (trigōnon) and measure (metron). Though the field emerged in Greece during the third century B.C., some of the most important contributions (such as the sine function) came from India in the fifth century A.D. Because early trigonometric works of Ancient Greece have been lost, it is not known whether Indian scholars developed trigonometry independently or after Greek influence. According to Victor Katz in “ A History of Mathematics (3rd Edition) ” (Pearson, 2008), trigonometry developed primarily from the needs of Greek and Indian astronomers. An example: Height of a sailboat mast Suppose you need to know the height of a sailboat mast, but are unable to climb it to measure. If the mast is perpendicular to the deck and top of the mast is rigged to the deck, then the mast, deck and rigging rope form a right triangle. If we know how far the rope is rigged from the mast, and the slant at which the rope meets the deck, then all we need to determine the mast’s height is trigonometry.

For this demonstration, we need to examine a couple ways of describing “slant.” First is slope, which is a ratio that compares how many units a line increases vertically (its rise) compared to how many units it increases horizontally (its run). Slope is therefore calculated as rise divided by run. Suppose we measure the rigging point as 30 feet (9.1 meters) from the base of the mast (the run). By multiplying the run by the slope, we would get the rise — the mast height. Unfortunately, we don’t know the slope. We can, however, find the angle of the rigging rope, and use it to find the slope. An angle is some portion of a full circle, which is defined as having 360 degrees. This is easily measured with a protractor. Let’s suppose the angle between the rigging rope and the deck is 71/360 of a circle, or 71 degrees. We want the slope, but all we have is the angle. What we need is a relationship that relates the two. This relationship is known as the “tangent function,” written as tan(x). The tangent of an angle gives its slope.

For our demo, the equation is: tan(71°) = 2.90. (We'll explain how we got that answer later.) This means the slope of our rigging rope is 2.90. Since the rigging point is 30 feet from the base of the mast, the mast must be 2.90 × 30 feet, or 87 feet tall. (It works the same in the metric system: 2.90 x 9.1 meters = 26.4 meters.) Sine, cosine and tangent Depending on what is known about various side lengths and angles of a right triangle, there are two other trigonometric functions that may be more useful: the “sine function” written as sin(x), and the “cosine function” written as cos(x). Before we explain those functions, some additional terminology is needed. Sides and angles that touch are described as adjacent. Every side has two adjacent angles. Sides and angles that don’t touch are described as opposite. For a right triangle, the side opposite to the right angle is called the hypotenuse (from Greek for “stretching under”). The two remaining sides are called legs.

Usually we are interested (as in the example above) in an angle other than the right angle. What we called “rise” in the above example is taken as length of the opposite leg to the angle of interest; likewise, the “run” is taken as the length of the adjacent leg. When applied to an angle measure, the three trigonometric functions produce the various combinations of ratios of side lengths. In other words: The tangent of angle A = the length of the opposite side divided by the length of the adjacent side The sine of angle A = the length of the opposite side divided by the length of the hypotenuse The cosine of angle A = the length of the adjacent side divided by the length of the hypotenuse From our ship-mast example before, the relationship between an angle and its tangent can be determined from its graph, shown below. The graphs of sine and cosine are included as well. The three principle trigonometric functions. Credit: Robert J. Coolman Worth mentioning, though beyond the scope of this article, is that these functions relate to each other through a great variety of intricate equations known as identities , equations that are always true.

Each trigonometric function also has an inverse that can be used to find an angle from a ratio of sides. The inverses of sin(x), cos(x), and tan(x), are arcsin(x), arccos(x) and arctan(x), respectively. The inverses of the three principle trigonometric functions. Credit: Robert J. Coolman Shapes other than right triangles Trigonometry isn’t limited to just right triangles. It can be used with all triangles and all shapes with straight sides, which are treated as a collection of triangles. For any triangle, across the six measures of sides and angles, if at least three are known the other three can usually be determined. Of the six configurations of three known sides and angles, only two of these configurations can’t be used to determine everything about a triangle: three known angles ( AAA ), and a known angle adjacent and opposite to the known sides ( ASS ).

Unknown side lengths and angles are determined using the following tools: The Law of Sines , which says that if both measures of one of the three opposing angle/side pairs are known, the others may be determined from just one known: sin(A)/a = sin(B)/b = sin(C)/c The Law of Cosines , which says that an unknown side can be found from two known sides and the angle between them. It’s essentially the Pythagorean Theorem with a correction factor for angles that aren’t 90 degrees: c2 = a2 + b2 – 2ab∙cos(C) The fact that all the angles in a triangle must add up to 180 degrees: A + B + C = 180° The history of trigonometry Trigonometry follows a similar path as algebra : it was developed in the ancient Middle East and through trade and immigration moved to Greece, India, medieval Arabia and finally Europe (where consequently, colonialism made it the version most people are taught today). The timeline of trigonometric discovery is complicated by the fact that India and Arabia continued to excel in the study for centuries after the passing of knowledge across cultural borders.

For example, Madhava’s 1400 discovery of the infinite series of sine was unknown to Europe up through Isaac Newton’s independent discovery in 1670. Due to these complications, we’ll focus exclusively on the discovery and passage of sine, cosine, and tangent. Beginning in the Middle East, seventh-century B.C. scholars of Neo-Babylonia determined a technique for computing the rise times of fixed stars on the zodiac. It takes approximately 10 days for a different fixed star to rise just before dawn, and there are three fixed stars in each of the 12 zodiacal signs; 10 × 12 × 3 = 360. The number 360 is close enough to the 365.24 days in a year but far more convenient to work with. Nearly identical divisions are found in the texts of other ancient civilizations, such as Egypt and the Indus Valley . According to Uta Merzbach in “ A History of Mathematics ” (Wiley, 2011), the adaptation of this Babylonian technique by Greek scholar Hypsicles of Alexandria around 150 B.C.

was likely the inspiration for Hipparchus of Nicea (190 to 120 B.C.) to begin the trend of cutting the circle into 360 degrees. Using geometry, Hipparchus determined trigonometric values (for a function no longer used) for increments of 7.5 degrees (a 48th of a circle). Ptolemy of Alexandria (A.D. 90 to 168), in his A.D. 148 “ Almagest ”, furthered the work of Hipparchus by determining trigonometric values for increments of 0.5 degrees (a 720th of a circle) from 0 to 180 degrees. The oldest record of the sine function comes from fifth-century India in the work of Aryabhata (476 to 550). Verse 1.12 of the “ Aryabhatiya ” (499), instead of representing angles in degrees, contains a list of sequential differences of sines of twenty-fourths of a right angle (increments of 3.75 degrees). This was the launching point for much of trigonometry for centuries to come.

The next group of great scholars to inherit trigonometry were from the Golden Age of Islam. Al-Ma'mun (813 to 833), the seventh caliph of the Abbasid Caliphate and creator of the House of Wisdom in Baghdad, sponsored the translation of Ptolemy’s "Almagest" and Aryabhata’s "Aryabhatiya" into Arabic. Soon after, Al-Khwārizmī (780 to 850) produced accurate sine and cosine tables in “Zīj al-Sindhind” (820). It is through this work that that knowledge of trigonometry first came to Europe. According to Gerald Toomer in the “ Dictionary of Scientific Biography 7 ,” while the original Arabic version has been lost, it was edited around 1000 by al-Majriti of Al-Andalus (modern Spain), who likely added tables of tangents before Adelard of Bath (in South England) translated it into Latin in 1126.

Additional resources Trigonometric Functions | Trig Function | Edurite Math Trigonometric Functions posted on: 31 May, 2012 | updated on: 18 Aug, 2012 The functions of angles are known as trigonometric functions. They are also considered as circular functions. They are helpful in building the relationship of angles of triangle to length of sides of the triangle. Sine, cosine, tangent are the functions with which we are more familiar. These Trigonometric Functions can be defined as the proportionate ratio of two sides of the right angle triangle. Trigonometric function have wide range of applications including the computation of lengths and angles of right triangle. They are useful in navigation, physics, astronomy and engineering. Basically there are six trigonometric functions they are sine, cosine, tangent, secant, cosecant, cotangent. Now we will define the trigonometric functions by taking the right triangle that contains the angle ‘θ’ (pronounced as theta) and three sides of triangle: The hypotenuse is the side which is opposite to the 900. This is the longest side of right triangle. The adjacent side is the side adjacent to hypotenuse and opposite side.

The opposite side is the side opposite to the angle Ө. The measurement of angles inside the triangle is of 1800. Therefore in the right triangle one angle measures 900 and rest of the two angles will be acute angles whose sum will be equal to 900. The trigonometric functions are summarized as shown below, where the angle ‘θ’ is the angle between the adjacent side and the hypotenuse. They are defined as: Sine function (sin) = opposite / hypotenuse, cosine function (cos) = adjacent / hypotenuse, tangent function (tan) = opposite / adjacent, cotangent function (cot) = adjacent / opposite, secant function ( sec) = hypotenuse / adjacent, cosecant function (cosec) = hypotenuse / opposite. cosec, sec, cot functions are the reciprocal function of sin, cos and tan respectively. The inverse functions of these trigonometric functions are: the inverse of sin function is arcsin. inverse of cos function is arcos. inverse of tan function is arctan. inverse of sec function is arcsec. inverse of cosec function is arccosec. inverse of cot function is arccot.

There exists a relationship between the sine and cosine function which can be explained with the help of Pythagorean trigonometric identity. cos2 θ + sin2 θ = 1. Topics Covered in Trigonometric Functions posted on: 19 Jul, 2012 | updated on: 20 Jul, 2012 Trigonometry is a branch of mathematics which deals with angles and sides of right angled triangle. There are six types of trigonometric ratios that are defined as Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. These all trigonometric ratios depend or may be determined by three parameters that are Perpendicular, Base and...Read More Trigonometry Table posted on: 19 Jul, 2012 | updated on: 20 Jul, 2012 Trigonometry is a branch of mathematics with help of which we can determine heights and distances using variables. It can possibly be of three types: core, plane, spherical and analytic. Before studying trigonometry table let us understand few terms. 1.

Core Trigonometry: This type of trigonometry is used for triangles having one angle as right ang...Read More Heights and Distances posted on: 21 Jul, 2012 | updated on: 25 Jul, 2012 When we come across problems of heights and distances, we find that we have problems where we have to either find height of the tower or distance of the tower from a given point or some times we need to find distance of top of building from any point on level of base of building. We observe that figure formed for study of such real life problems is a...Read More Periodic Functions posted on: 18 Aug, 2012 | updated on: 23 Aug, 2012 Function that repeats the same behavior after a certain interval (period of function) is called as a periodic function. The examples of periodic functions I maths can be the six trigonometric Functions and also their inverse trigonometric functions.

A very important distinctive thing about functions is that they possess single output for each and every inpu...Read More Evaluation of Trigonometric Functions posted on: 18 Aug, 2012 | updated on: 23 Aug, 2012 Trigonometry is defined as a branch of mathematics which deals with angles and triangles or we can say that, it deals with the right angle geometry. A right angle triangle consists of three sides that are specifically termed as perpendicular, base and height. Trigonometry consists of six trigonometric functions that are, Sine function...Read More posted on: 13 Jul, 2012 | updated on: 13 Jul, 2012 The table of trigonometric angles is shown below. X TechnologyUK - Mathematics - Trigonometry - The Cosecant Function The cosecant function Trigonometric identities The Cosecant Function The cosecant function is a trigonometric function, one of three reciprocal functions that we look at in these pages, the other two being the secant function and the cotangent function. A reciprocal function is one that is the reciprocal (or multiplicative inverse) of another function (see below).

The cosecant function (usually abbreviated as csc) is the reciprocal function of the sine function. As with the other trigonometric functions in modern use, the cosecant function was certainly known to Arab scholars, and tables of cosecant values are known to have been in existence by the tenth century CE. In order to clarify exactly how the cosecant function and the sine function are related, consider the diagram below. The cosecant as a line segment of the unit circle Like the other trigonometric functions, the cosecant can be represented as a line segment associated with the unit circle. The diagram shows the cosecant for an angle of rotation θ of forty-five degrees (measured anti-clockwise from the positive x-axis). Line segment AF lies on the line that is tangent to the circle at point A. Line segment OF (shown in red) represents the cosecant, and is an extension of line segment OP. Remember that point P is a point on the circumference of the circle whose x and y coordinates represent the value of cos (θ ) and sin (θ ) respectively (the line segments representing the sine and cosine are also shown).

Click here to see an interactive demonstration that uses the unit circle to show how the sine and the cosecant functions relate to one another (note: your browser must support Java in order for the interactive page to work). If you haven't already deduced the fact from the diagram, you will see from the interactive demo that as the value of sin (θ ) approaches one, so does the value of csc (θ ). As the value of sin (θ ) approaches zero, however, the value of csc (θ ) tends to infinity. The value of csc (θ ) when sin (θ ) equals zero is thus said to be undefined. Obviously, since the cosecant function is the reciprocal of the sine function, it can be expressed in terms of the sine function as: csc (θ ) = 1 sin (θ ) Since the sine is represented by the y coordinate of point P (i.e. the point at which the terminal side of angle θ intersects the circumference of the unit circle), we can also conclude that: csc (θ ) = 1 y We can also define the cosecant in terms of a right-angled triangle.

If you have read the pages in this section covering the three main trigonometric functions (sine, cosine and tangent) you will by now be familiar with the diagram below. In triangle ABC, csc (θ ) = c/a In right-angled triangle ABC, the sine of angle θ is the quotient of the opposite and the hypotenuse (i.e. the sides labelled a and c respectively). Since the cosecant function is the reciprocal of the sine function, then the cosecant of angle θ will be the quotient of the hypotenuse and the opposite. This gives us: csc (θ ) = opposite a For those interested in such things, we have used Microsoft Excel to produce our own table of cosecant values for angles ranging from zero degrees (0°) up to three hundred and sixty degrees (360°) in increments of one tenth of a degree. To see the table, click here . If you study the table, you can see that there is a significant range of angles centred around ninety degrees for which the value returned by the cosecant function is quite small and changes relatively slowly.

For angles close to either zero degrees (0°) or one hundred and eighty degrees (180°) however, the value returned by the cosecant function is much bigger. This is to be expected, since the value of sin (θ ) will be very small for these angles, so its reciprocal will be correspondingly large. In fact, the value returned by the cosecant function for an angle of either zero degrees or one hundred and eighty degrees is considered to be undefined, since the equation csc (θ ) = 1/sin (θ ) will involve division by zero. The same is true for an angle of three hundred and sixty degrees (360°). In fact, it will apply to any angle for which the sine value is zero. We present below the graph of the cosecant function for angles in the range zero degrees to seven hundred and twenty degrees (720°). A graph of the cosecant function for angles in the range 0° to 720° Because the absolute value of the sine of an angle will never exceed one, the reciprocal of the sine of an angle will never be less than one.

So, just as the cosecant value is undefined for any angle for which the sine is zero, it will always be one for any angle for which the sine is one, and minus one (-1) for any angle for which the sine is minus one. In fact, as you can see from both the table of cosecant values and the graph of the cosecant function, the cosecant value is negative for all angles for which the sine is negative. Like the graph of the tangent function, the graph of the cosecant function can include some very large positive and negative peaks, as the value of sin (θ ) tends to zero. Unlike the tangent function however, the period of the cosecant function is three hundred and sixty degrees (360°), or 2π. Like the sine function, the cosecant function returns positive values for angles in Quadrants I and II, and negative values for angles in Quadrants III and IV. If you have read the page entitled "The Tangent Function" (or possibly even if you haven't) you will know that the vertical red lines on the graph are asymptotes.

The asymptote of a curve is a line such that, as the distance between the curve and the line approaches zero, the value of the curve tends to infinity. In this case, the asymptotes can be considered to be the "gaps" in the graph where the value of csc (θ ) is undefined. Notice also that as the graph crosses the asymptote, the value of csc (θ ) transitions from its maximum positive value to its maximum negative value (or vice versa). You can also see how the values differ on either side of the asymptote by studying the table of cosecant values. As with the other trigonometric functions we have looked at, we don't need to use tables to find the cosecant of an angle (or the angle corresponding to a given cosecant value), since any modern scientific calculator can do it for us, even though most calculators don't have a dedicated button for the cosecant function. Finding the cosecant of an angle with the built-in calculator provided in Microsoft Windows is still relatively straightforward. Let's assume we want to find the cosecant of an angle of eighty-five degrees (85°).

You can find the Microsoft Windows built-in calculator by clicking on the start button in Windows 7 and selecting All Programs>Accessories>Calculator (other versions of Windows might require a different sequence of keystrokes). The scientific version of the calculator can be selected from the application's View menu. To find the cosecant of eighty-five degrees using the Windows calculator, enter the following keystrokes (if you don't have the Windows calculator, use any available calculator that can perform trig functions): Enter these keystrokes to find the cosecant of 85° Since the cosecant function is the reciprocal of the sine function, all we have done here is to use the sine function button on the calculator, and then apply the calculator's built-in reciprocal function button (usually labelled 1/x) to the result. Make sure, by the way, that you have the calculator set to the Degrees mode (unless you are planning to enter a value for the angle in radians or gradians). If you have entered the keystrokes correctly, you should now see the following display: The calculator displays the value of csc(85°) So far, so good.

Now let's assume that you want to find the size of an acute angle in a right-angled triangle by first finding the quotient of the hypotenuse and opposite (i.e. the angle's cosecant). Providing you have values for the lengths of the sides of the triangle, you can find the value of the angle's cosecant easily enough. Finding the angle is not quite as straightforward as for the three main trigonometric functions, as most calculators do not have a dedicated button for the arccosecant function (which is the inverse of the cosecant function). Fortunately, there is a relatively simple way around that. Let's find the value of angle θ for the right-angled triangle illustrated below. We want to find the size of angle θ For angle θ, sides c and a are the hypotenuse and the opposite respectively. We can actually find the size of the angle in a single operation on the Windows calculator. We do this by using brackets in the key sequence entered, in order to get the calculator to find the inverse of the quotient of the hypotenuse and opposite (i.e. the cosecant of the angle) first. We then apply the arcsine function to the result.

Here is the key sequence to use: Enter these keystrokes to find the size of angle θ You may have already worked out that what we did here was to find the sine of the angle by taking the reciprocal of the cosecant. We then simply applied the arcsine function to the result. We could of course simply reverse the order in which we enter the values (i.e. opposite / hypotenuse), in which case we don't need to use the reciprocal button. What should start to emerge for you at this point is the fact that all of the trigonometric functions are very closely interrelated, and that we can often replace an expression involving one trigonometric function with an equivalent expression involving a different trigonometric function. This can sometimes be very useful. If you have entered the keystrokes correctly, you should now see the following display: The calculator displays the value of angle θ The cosecant function, in common with other trigonometric functions, can be represented as an infinite series. As we have mentioned elsewhere, the value returned by a trigonometric function for a given angle cannot be calculated using a simple algebraic formula.

For most angles, an exact value can only be obtained as the sum of an infinite number of terms, which is obviously impossible to achieve. A reasonably accurate approximation of the value is found using the sum of a limited number of terms from the function's infinite series. The greater the number of terms included, the greater the accuracy of the result. The number of terms used will depend on the accuracy required, and on the computational resources available. The infinite series frequently cited for the cosecant function looks like this: csc (x) = x-1 + TRIGONOMETRY TRIGONOMETRY Freebies TRIGONOMETRY Cosine, Sine and Tangent are three of a series of mathematical functions used in the study of right angled triangles (any triangle where one angle is at 90 degrees (see figure 1). This is part of the branch of maths known as Trigonometry - the study of all triangles. These three functions represent the ratio between two of the sides of a right angled triangle for a particular angle. For any particular angle, this ratio will always be the same, regardless of the actual size of the triangle.

If you know the angle of one other corner of the triangle, and the length of any one side of the triangle, you can calculate the length of the other two sides. There are several different ways of measuring angles, of which degrees are the most familiar, and will be used here. Poser uses the Radian, which is much more elegant mathematically. The basic principles of trigonometry are the same regardless of which method you use to measure angles. Fig 1: The Parts of the Triangle The Parts of the Triangle To understand what's going on in these functions, we need to have clear names for each part of the triangle. Figure 1 and the list below shows the four most important terms used here. b = the angle h = hypotenuse: the long side, opposite the right angle o = opposite: the side opposite to the angle 'b' a = adjacent: the short side next to the angle 'b' Sine, Cosine and Tangent For any given angle of b, we can calculate the sine, cosine and tangent by measuring the sides of the triangle and performing the following calculations.

For our example, we will use a triangle where a = 1, o = 2, h = 2.236 and b = 63.434 degrees. sin (b) = o / h sin (63.434) = 2 / 2.236 = 0.894 cos (b) = a / h cos (63.434) = 1 / 2.236 = 0.447 tan (b) = o / a tan (63.434) = 2 / 1 = 2 It doesn't matter how big the triangle is - as long as the angle b stays the same then the ratio between the sides of the triangle will remain the same, and the results of the sine, cosine and tangent calculations will remain the same (for example if a=1.5, then o=3, resulting in tan(63.434)= 3 / 1.5 = 2). Although there are mathematical ways to calculate the values of sine, cosine and tangent just from the angle, they are complex and time consuming and we don't need to use them.

Poser provides us with maths functions that do the work for us (see below for how the trig. functions were used in the past). Derived Functions Now we have these three functions, we need to modify them so that we can use them when we know 'b' and one other value. We can do that using some simple mathematical tricks. When we have a formula like sin (b) = o / h we can perform maths on the formula as long as we do the same thing to both sides of the formula. First of all, we will multiply both sides by h. That will turn sin (b) into sin (b) * h. On the other side, o / h is turned into just o (if you take any number, divide it by another number and then multiply it by the same number you will always return to the original number). This gives us a new formula: sin (b) * h = o. Finally, swap the sides around, and we get o = sin (b) * h. If we know the angle and the hypotenuse we can use this new formula to work out the opposite. We can go one step further.

o = sin (b) * h can also be written as o = h * sin (b). If we divide both sides of this formula by sin (b) we end up with o / sin (b) = h. Swap that around and we get h = o / sin (b). Table 1 shows all six formula that can be created in this way. TABLE 1: Sine, Cosine and Tangent formula. sin (b) = o / h Table 2 shows which formula to use in each possible set of circumstances. TABLE 2: Which formula to use when - Using Sine, Cosine or Tangent The standard example used to explain when you might want to use trigonometry is the measurement of heights. Let us say that you need to know the height of the top of your house. Actually measuring up the side of the building would be awkward to say the least. Instead, we will use our trigonometry. First, pick a spot a little way away from the building. Measure the distance from that spot to the base of the house wall. This will be our 'a' or adjacent. In this example, we ended up 10 meters away from the house.

Second, from your spot measure the angle between the ground and the highest point on the building. This becomes our 'b' or angle. In this example, the angle from our spot 10 meters away from the house to tip of the roof was 30 degrees. We now have a and b, and we need to calculate 'o', the opposite. If you look at the list of functions we have above, you will see that o = tan (b) * a. That means that 0 = tan (30) * 10. Tan (30) is equal to 0.577. Therefore our house is 0.577 * 10, or 5.77 meters high. Calculating Sine, Cosine or Tangent. Prior to the advent of the scientific calculator and the computer, if you needed to find out a value for sine, cosine or tangent, you would use a lookup table. This was simply a pre-calculated list of the values of the trigonometry functions for a particular set of angles. If you needed an angle that isn't on the list, you could estimate it from the list.

On our sample table below, we have values for 1 and 2 degrees. If we wanted sine(1.5), first we would look at the values for 1 and 2 and calculate the gap between them (0.0349-0.0175= 0.0174). Next, we would look at the fraction of our angle (.5), and multiply our 0.0175 by 0.5, to get 0.0087. Finally, we would add that to the value for 1 degree, reaching an approximation of 0.0175+0.0087= 0.0262. This is actually accurate to the four decimal places we see - the windows Calculator returns sine(1.5)= 0.026176948307873152610611685554113 - if we round this up to 4 decimal places, then the final result is 0.262.

If you were working in a field that needed more accuracy that this (aircraft design for instance), then you'd buy a more detailed set of lookup table. Radians

Question: Sine, cosine and tangent are the three most familiar functions in which branch of mathematics?
Answer: Trigonometry